On this page (which has not been updated in a while…) we illustrate some of our work through animations to complement our technical journal papers.

We model **compositional multiphase flow in porous media** with higher-order finite element (FE) methods. In traditional, lowest-order, finite difference (FD) numerical methods, discretized variables are *constant* in each grid element, while higher-order methods use linear or *higher-order polynomial approximations*. The use of an element-wise constant approximation (FD) leads to a large numerical error, which is referred to as *numerical dispersion*. This error can be reduced by using extremely fine grids, but this is computationally very expensive. By using higher-order methods, the numerical dispersion can be reduced considerably, and accurate simulations can be performed on coarse grids at orders of magnitude higher CPU efficiency.

The ‘smearing out’ of sharp features by numerical dispersion has the dangerous effect of artificially stabilizing important flow instabilities, such as **gravitational and viscous fingering**. Gravitational fingering occurs when a denser fluid resides on top of a lighter fluid, or a lighter fluid underneath a denser one. This can easily occur when (supercritical) CO_{2} is injected in hydrocarbon reservoirs or saline aquifers. Viscous fingering is a similar instability, which can reduce sweep efficiency when a less viscous fluid (typically gas) is injected in a more viscous one (oil/water). These instabilities have a profound impact on flow paths, but the onset can easily be suppressed by numerical dispersion when these processes are simulated with a FD approach on relatively coarse grids (which is often inevitable for large reservoirs). The panels below show simulation results for supercritical CO_{2} injection in the top of a core saturated with lighter oil. In the DG simulation (left) we capture the resulting gravitational fingering instability, while a lowest-order FD simulation (right) predicts piston-like displacement, because the instability is stabilized by numerical dispersion, even on this relatively fine (18 by 18 by 80 element) grid. More details are provided in Moortgat et al., *SPE J* (2013) 18(2), 331-344 (click on Fig. for pdf).

Gravitational fingering also occurs in large scale reservoirs, and not only when a dense fluid is injected on top of a lighter one. The figure at the top of this page is for a reservoir where CO_{2}, injected from the top, dissolves in the oil, which increases the oil density locally in the top and triggers profound fingering behavior throughout the reservoir. The instability is hard to avoid in this case, and a different recovery strategy may be more effective for this field.

The animation below shows viscous fingering occurring in water-alternating-gas (WAG) injection, when the gas slugs have a much lower viscosity than the reservoir oil.

**Fractured reservoirs ** are particularly challenging, due to the orders of magnitude ranges in spatial scales (from sub-mm wide fractures to the km^{3} reservoir scale) and rock properties (fracture permeability can easily be 10^{8} times the *matrix* permeability). The most common fracture models are the single- and dual-porosity (/permeability) models. The former discretizes fractures explicitly, which is physically robust but extremely CPU expensive, while the latter is very efficient but makes unreasonable approximations for the fracture-matrix interactions. The dual-porosity approach assumes that all the flow is through a sugar-cube configuration of fractures (modeled on one grid) and that only diffusive flow occurs in the matrix blocks (modeled on a second grid), which serve mainly as oil storage. The sugar-cube fracture configuration is an unreasonable representation of real fractured formations. We have developed an alternative approach that combines the benefits of both the single- and dual-porosity models, while avoiding their limitations. This *cross-flow equilibrium model* allows for any configuration of *discrete fractures*, explicitly computes all fracture-matrix interactions and the (convective) fluxes in matrix blocks when important, at high CPU efficiency.

**Fickian diffusion** and **capillarity** complicate the modeling of compositional multiphase flow in general, and in fractured media in particular. The animations below show the detrimental effect of capillarity on gravity drainage from a (discrete) fractured oil reservoir. The left panel shows the gas saturation when capillarity is neglected, and the right panel takes capillarity into consideration. This results are from Moortgat & Firoozabadi, *SPEJ* (2013), doi:10.2118/159777-PA (click on Fig. for pdf).

The modeling of **Fickian diffusion in fractured media** has been plagued by a severe numerical issue: when fractures quickly fill with injected gas, while the matrix is still fully saturated with oil, one cannot compute a diffusive flux from the *phase* compositions in neighboring cells. As a result, the beneficial effect of diffusion has been significantly underestimated for gas injection in fractured oil reservoirs. We have overcome this issue by reformulating the driving force for Fickian diffusion in terms of *chemical potential* gradients, rather than gradients in phase compositions. The chemical potential of a species in a single- or multiphase mixture is the same in all phases, and gradients in chemical potential can always be rigorously defined between neighboring elements. The panels below show the gas saturation throughout a densely fractured reservoir during CO_{2} injection. When Fickian diffusion is neglected, or computed from compositional gradients, injected CO2 quickly flows through the fractures and results in early breakthrough and low oil recovery (left panel). Diffusion drives considerable species exchange across the large interaction surface between fractures and matrix blocks (right panel). Additionally, when CO_{2} dissolves in the matrix oil, it can swell the oil and expel it from the matrix, and reduce the oil viscosity. Moortgat & Firoozabadi, Energy & Fuels (2013), 27(10),5793–5805 (click on Fig. for pdf).

**Unstructured grids** are pretty much unavoidable to reliable mesh the complex geometries found in typical geological formations. Finite element methods are a natural choice to model flow on such unstructured grids. Our reservoir simulator has the flexibility to use any combination of hexahedral, prismatic, and tetrahedral grid elements, which allows us to easily conform to reservoirs of any complexity, as illustrated below (3D paper in review; 2D unstructured grids in Moortgat & Firoozabadi, *AWR* (2010) 33, 951-968).

**Phase behavior** of two- and three-phase hydrocarbon mixtures or mixtures containing polar molecules, such as water and asphaltenes, are another important aspect of our work.

We have developed highly robust, accurate, and efficient (three-)phase-split calculations that 1) are based on accurate equations of state (EOS), 2) correspond to the global minimum of Gibbs free energy, verified by phase stability analyses, and 3) guarantee the equality of chemical potentials (/fugacities) of each species in all the phases. The latter two conditions are required to satisfy *local* thermodynamic equilibrium.

We use the standard Peng-Robinson EOS for pure hydrocarbon phases, but have also developed a more general cubic-plus-association EOS to take into account the self- and cross-association due to polar water and asphaltene molecules. Moortgat, Li, and Firoozabadi, *WRR* (2012) 48, W12511.